When encountering abstract algebra for the first time, students react in one of two ways: either they are intrigued by its astonishing beauty and its elegant proofs, or they struggle with the vast amount of dry, complicated definitions that are far from intuitively clear and are, not surprisingly, awfully abstract. This textbook offers a little bit for both types of reader.

The author promises an "introduction to groups, rings and fields" and it is important to bear this in mind. Although Abstract Algebra is aimed at second- to fourth-year undergraduates, Clive Reis devotes about 100 pages to familiarising the reader with the most basic logic, set theory and algebraic properties of the integers and their congruence classes modulo n, which enables him to develop most concepts of group theory in clear analogy to the ideas already discussed. Although some students may find that a stronger emphasis on explaining more advanced topics such as field extensions would have been more helpful than another detailed discussion of the principle of induction, most will appreciate the way the material is made accessible: both examples and exercises (which are often closely related to the text, but may also convey other central notions) are well chosen. Important results appear in the most natural order and are often developed step by step rather than merely presented.

What makes this book more than just a safe journey from square one to the usual results such as Lagrange's theorem and the Sylow theorems is the discussion of several applications, illustrating both the amazing power of these concepts and the diversity of fields (no pun intended) where abstract algebra can prove helpful: Latin squares, Polya-Burnside enumeration and isometries in Euclidean space. By the time we reach error-correcting code, our type II students will have made their peace with the subject and gained some insight into the relevant concepts of group theory, and type I students will be smiling broadly, ready to forgive the initial focus on basics.

Who is it for? Mathematics undergraduates.

Presentation: Generic, with some rather colloquial side remarks. Some typographic mistakes ("ipimorphism", unbalanced brackets) will need to be corrected in future editions.

Would you recommend it? Yes, but different chapters to different people.